In order to use Proposition 5.5, we need to show the conditions in Lemma 5.3 in [1]} holds for any compact subset of the parameter space. That is, for any \(c>0\)  and sequence \(\lbrace \mathbf {y}_i\rbrace _{i=1}^{\infty }\)  satisfying \(\Vert \mathbf {y}_i\Vert \le c\) , the sequence of functions \({\psi _i(\mathbf {y}_i,\mathbf {\theta })}\)  is equicontinuous on any compact set of the parameter space. It is easy to see that \(\dfrac{\partial \psi _i}{\partial \mathbf {\theta }^T}\)  (derived in the previous subsection) is uniformly bounded in any compact subset \(\Theta \)  of the parameter space when \(\Vert \mathbf {y}_i\Vert \le c\)  if \(\sup _i\Vert \mathbf {X}_i\Vert <\infty \) , \(\sup _i\Vert \mathbf {Z}_i\Vert  < \infty \) , \(\sup _i J_i <\infty \)  and \(\sup _i\Vert \mathbf {\Sigma }_i^{-1}\Vert _\infty <\infty \) , where \(\Vert \cdot \Vert _\infty \)  indicates matrix infinity norm. Since \(\sup _i\Vert \mathbf {\Sigma }_i^{-1}\Vert _\infty <\infty \)  if and only if \(\sup _i\Vert \mathbf {Z}_i\Vert <\infty \)  and \(\inf _i\det (\mathbf {\Sigma }_i)>0\) , the aforementioned conditions holds under the regularity conditions. Therefore, \(\psi _i(\mathbf {y}_i,\mathbf {\theta })\)  is equicontinuous on \(\Theta \) . Moreover, since \(\mathbf {Y}_i\)  has finite \((4+\delta )\) -th moment, \(\mathbb {E}\lbrace \sup _{\mathbf {\theta }\in \Theta }\Vert \psi _i(\mathbf {Y}_i,\mathbf {\theta })\Vert \rbrace ^{2}<\infty \)  and \(\sup _i\mathbb {E}\Vert \mathbf {Y}_i\Vert <\infty \) , the conditions in Lemma 5.3 in [1]} holds.
